On the cp-rank and minimal cp factorizations of a completely positive matrix

We show that the maximal cp-rank of $n\times n$ completely positive matrices is attained at a positive-definite matrix on the boundary of the cone of $n\times n$ completely positive matrices, thus answering a long standing question. We also show that the maximal cp-rank of $5\times 5$ matrices equals six, which proves the famous Drew-Johnson-Loewy conjecture (1994) for matrices of this order. In addition we present a simple scheme for generating completely positive matrices of high cp-rank and investigate the structure of a minimal cp factorization.


appeared in SIAM J. Matrix Analysis Appl., http://epubs.siam.org/doi/abs/10.1137/120885759